The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(1)).

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

f(n__b, X, n__c) → f(X, c, X)
c → b
b → n__b
c → n__c
activate(n__b) → b
activate(n__c) → c
activate(X) → X

Rewrite Strategy: INNERMOST

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(n__b, z0, n__c) → f(z0, c, z0)
c → b
c → n__c
b → n__b
activate(n__b) → b
activate(n__c) → c
activate(z0) → z0

Tuples:

F(n__b, z0, n__c) → c1(F(z0, c, z0), C)
C → c2(B)
ACTIVATE(n__b) → c5(B)
ACTIVATE(n__c) → c6(C)

S tuples:

F(n__b, z0, n__c) → c1(F(z0, c, z0), C)
C → c2(B)
ACTIVATE(n__b) → c5(B)
ACTIVATE(n__c) → c6(C)

K tuples:none
Defined Rule Symbols:

f, c, b, activate

Defined Pair Symbols:

F, C, ACTIVATE

Compound Symbols:

c1, c2, c5, c6

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 4 trailing nodes:

ACTIVATE(n__c) → c6(C)
C → c2(B)
F(n__b, z0, n__c) → c1(F(z0, c, z0), C)
ACTIVATE(n__b) → c5(B)

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(n__b, z0, n__c) → f(z0, c, z0)
c → b
c → n__c
b → n__b
activate(n__b) → b
activate(n__c) → c
activate(z0) → z0

Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

f, c, b, activate

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(6) BOUNDS(O(1), O(1))